Subgroups — C₂×M₁₆×#1057 #16792567

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719119	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#1057	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#1057	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718251	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#1057	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#1057	C₄	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊#1057	C₄	1	2	yes	—
12	#11719119	C₄	1	4	yes	—
13	#11719119	C₂²	1	4	yes	—
14	C₂×C₄⋊#1057	C₄	1	2	yes	—
15	C₂×C₄×#1057	C₄	1	4	yes	—
16	C₂×C₄×#1057	C₄	1	2	yes	—
17	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
18	C₂×C₄×#1057	C₂²	1	2	yes	—
19	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1057	C₂²	1	4	yes	—
21	M₁₆×Dic₁₃	C₄	1	4	yes	—
22	M₁₆×Dic₁₃	C₂²	1	4	yes	—
23	C₂×#88055	C₄	1	2	yes	—
24	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
25	#11718251	C₂²	1	8	yes	complemented
26	C₈×#1057	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1057	—	2	8	no	—
31	C₂²×#1057	C₂×C₄	1	4	yes	—
32	C₂²×#1057	C₂×C₄	1	2	yes	—
33	C₂²×#1057	C₂×C₄	1	8	yes	—
34	C₂²×#1057	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×M₁₆	C₈	1	8	yes	complemented
36	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂×#6099	C₈	1	8	yes	complemented
47	C₂×#6099	C₂×C₄	1	4	yes	—
48	C₂²×#1055	C₂×C₄	1	2	yes	—
49	C₂²⋊#1057	C₂×C₄	1	8	yes	—
50	C₄⋊#1057	C₂×C₄	1	8	yes	—
51	C₄×#1057	C₂×C₄	1	2	yes	—
52	C₄×#1057	C₂×C₄	1	8	yes	—
53	C₄×#1057	C₂×C₄	1	2	yes	—
54	C₄×#1057	C₂×C₄	1	4	yes	—
55	C₄×#1057	C₂³	1	2	yes	—
56	C₄×#1057	C₂³	1	4	yes	—
57	C₄×#1057	C₂³	1	2	yes	—
58	#88055	C₂×C₄	1	8	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
60	C₈×Dic₁₃	C₂³	1	4	yes	—
61	C₂×C₈×M₁₆	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₄²	1	2	yes	—
74	C₂²×Dic₁₃	C₄²	1	2	yes	—
75	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×#1057	—	2	4	no	—
80	C₂×#1057	—	2	8	no	—
81	C₂×#1057	—	2	16	no	—
82	C₂×#1057	M₁₆	1	8	yes	direct product factor
83	C₂×#1057	C₂²×C₄	1	4	yes	—
84	C₂×#1057	C₂²×C₄	1	8	yes	—
85	C₂×#1057	C₂²×C₄	1	4	yes	—
86	C₁₃×M₁₆	C₂×C₈	1	16	yes	complemented
87	C₂×C₈×C₁₃	C₂×C₈	1	8	yes	complemented
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₄×Dic₁₃	C₄²	1	2	yes	—
90	C₄×Dic₁₃	C₄²	1	2	yes	—
91	C₄×Dic₁₃	C₄²	1	2	yes	—
92	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
99	#6099	C₂×C₈	1	16	yes	complemented
100	C₂×#1055	C₂×C₈	1	8	yes	complemented
101	C₂×#1055	C₂²×C₄	1	4	yes	—
102	C₂×#431	—	13	1	no	—
103	C₂×C₄×M₁₆	—	13	1	no	—
104	C₂²×C₄×C₈	—	13	1	no	—
105	C₈×M₁₆	—	13	8	no	—
106	C₂×#322	—	13	2	no	—
107	C₂×C₈²	—	13	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₄×C₈	1	4	yes	—
111	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
114	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	2	4	no	—
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
129	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
130	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
133	C₂×Dic₁₃	#131	1	4	yes	—
134	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
136	#1057	—	2	4	no	—
137	#1057	—	2	8	no	—
138	#1057	C₂×M₁₆	1	8	yes	direct product factor
139	C₈×C₁₃	C₂²×C₈	1	8	yes	complemented
140	#1055	C₂²×C₈	1	8	yes	complemented
141	C₂²×M₁₆	—	13	1	no	—
142	C₂²×M₁₆	#261	1	1	yes	characteristic
143	C₂³×C₈	—	13	2	no	—
144	C₂²×C₄²	—	13	1	no	—
145	#431	—	13	4	no	—
146	#431	—	13	4	no	—
147	C₂×C₄⋊C₈	—	13	2	no	—
148	C₂×C₂²⋊C₈	—	13	2	no	—
149	C₄×M₁₆	—	13	4	no	—
150	C₄×M₁₆	—	13	4	no	—
151	C₂×#97	—	13	2	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	4	no	—
155	C₂×C₄×C₈	—	13	2	no	—
156	C₂×C₄×C₈	—	13	4	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	13	2	no	—
159	#322	—	13	8	no	—
160	C₈²	—	13	8	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	8	no	—
164	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
165	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
166	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
167	C₂²×C₁₃	#405	1	2	yes	—
168	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
169	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
170	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
171	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
173	Dic₁₃	—	2	2	no	—
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	C₄×M₁₆	1	2	yes	—
176	Dic₁₃	#405	1	2	yes	—
177	Dic₁₃	C₂²×M₁₆	1	2	yes	—
178	Dic₁₃	C₂×#131	1	2	yes	—
179	C₂³×C₄	—	13	1	no	—
180	C₂³×C₄	—	13	1	no	—
181	C₂³×C₄	C₂×#261	1	1	yes	characteristic
182	C₂×M₁₆	—	13	8	no	—
183	C₂×M₁₆	—	13	4	no	—
184	C₂×M₁₆	#1057	1	8	yes	direct product factor
185	C₂×M₁₆	C₂×#261	1	4	yes	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	8	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	4	no	—
191	C₂²×C₈	—	26	8	no	—
192	C₂²×C₈	C₂×#261	1	2	yes	—
193	C₂×C₄²	—	13	1	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	13	2	no	—
198	C₂×C₄²	—	13	2	no	—
199	C₂×C₄²	—	13	1	no	—
200	C₂×C₄²	—	13	1	no	—
201	C₄⋊C₈	—	13	8	no	—
202	C₂²⋊C₈	—	13	8	no	—
203	#97	—	13	8	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	8	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	—	13	4	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	4	no	—
210	C₄×C₈	—	13	2	no	—
211	C₄×C₈	—	13	4	no	—
212	C₄×C₈	—	13	4	no	—
213	C₂×C₁₃	—	2	4	no	—
214	C₂×C₁₃	C₈×M₁₆	1	4	yes	complemented
215	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
216	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
218	C₂⁴	C₄×#261	1	1	yes	characteristic
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	1	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	26	4	no	—
228	C₂²×C₄	—	26	4	no	—
229	C₂²×C₄	D₁₃⋊C₈	1	4	yes	—
230	C₂²×C₄	C₄×#261	1	1	yes	characteristic
231	C₂²×C₄	C₄×#261	1	2	yes	—
232	C₂²×C₄	C₂×#1057	1	4	yes	—
233	C₂²×C₄	C₂²×#261	1	2	yes	—
234	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
235	M₁₆	—	13	16	no	—
236	M₁₆	C₂×#1057	1	16	yes	direct product factor
237	C₂×C₈	—	13	4	no	—
238	C₂×C₈	—	13	8	no	—
239	C₂×C₈	—	13	8	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	8	no	—
243	C₂×C₈	—	26	16	no	—
244	C₂×C₈	—	26	4	no	—
245	C₂×C₈	—	26	8	no	—
246	C₂×C₈	C₂×#1057	1	8	yes	complemented
247	C₂×C₈	C₂²×#261	1	4	yes	—
248	C₄²	—	13	1	no	—
249	C₄²	—	13	1	no	—
250	C₄²	—	13	2	no	—
251	C₄²	—	13	2	no	—
252	C₄²	—	13	2	no	—
253	C₄²	—	13	2	no	—
254	C₄²	—	13	2	no	—
255	C₄²	—	13	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	13	2	no	—
258	C₁₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×#1057	1	4	yes	—
262	C₂³	C₂×C₄×#261	1	2	yes	—
263	C₂³	C₂×C₄×#261	1	1	yes	characteristic
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	13	4	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	1	no	—
268	C₂×C₄	—	13	1	no	—
269	C₂×C₄	—	13	1	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	26	4	no	—
276	C₂×C₄	—	26	2	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	C₄×#1057	1	8	yes	—
279	C₂×C₄	C₄×#1057	1	4	yes	—
280	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
281	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
282	C₂×C₄	C₂×C₄×#261	1	2	yes	—
283	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
284	C₂×C₄	C₂²×#1057	1	2	yes	—
285	C₂×C₄	C₂²×#1057	1	4	yes	—
286	C₂×C₄	C₂³×#261	1	1	yes	characteristic
287	C₈	—	13	8	no	—
288	C₈	—	13	8	no	—
289	C₈	—	26	4	no	—
290	C₈	—	26	8	no	—
291	C₈	C₂²×#1057	1	8	yes	complemented
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	M₁₆×#261	1	2	yes	—
296	C₂²	#11719099	1	2	yes	—
297	C₂²	C₂×C₄×#1057	1	4	yes	—
298	C₂²	C₂×C₄×#1057	1	2	yes	—
299	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
300	C₄	—	13	2	no	—
301	C₄	—	13	2	no	—
302	C₄	—	13	2	no	—
303	C₄	—	13	2	no	—
304	C₄	—	26	2	no	—
305	C₄	—	26	2	no	—
306	C₄	C₂×C₄×#1057	1	2	yes	—
307	C₄	C₂×C₄×#1057	1	4	yes	—
308	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
309	C₄	C₂³×#1057	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	M₁₆×#1057	1	4	yes	direct product factor
312	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
313	C₂	C₂×M₁₆×#261	1	1	yes	characteristic
314	C₂	C₂×#11719099	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. M₁₆×#1057	C₂	direct (N × H)
4. C₂×#11718251	C₂	semidirect
5. C₂×C₈×#1057	C₂	semidirect
25. #11718251	C₂²	semidirect
26. C₈×#1057	C₂²	semidirect
35. C₂×C₁₃×M₁₆	C₈	semidirect
46. C₂×#6099	C₈	semidirect
82. C₂×#1057	M₁₆	direct (N × H)
86. C₁₃×M₁₆	C₂×C₈	semidirect
87. C₂×C₈×C₁₃	C₂×C₈	semidirect
99. #6099	C₂×C₈	semidirect
100. C₂×#1055	C₂×C₈	semidirect
138. #1057	C₂×M₁₆	direct (N × H)
139. C₈×C₁₃	C₂²×C₈	semidirect
140. #1055	C₂²×C₈	semidirect
184. C₂×M₁₆	#1057	direct (N × H)
214. C₂×C₁₃	C₈×M₁₆	semidirect
236. M₁₆	C₂×#1057	direct (N × H)
246. C₂×C₈	C₂×#1057	semidirect
258. C₁₃	C₂×C₈×M₁₆	semidirect
291. C₈	C₂²×#1057	semidirect
311. C₂	M₁₆×#1057	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11719119	2
C₂²×C₄×#1057	2
M₁₆×#1057	2
C₂×#11718251	2
C₂×C₈×#1057	2
C₂×M₁₆×Dic₁₃	2
C₂×C₈×M₁₆	13

Subgroups — C2×M16×#1057 #16792567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×#1057 #16792567